Subalgebra \(A^{56}_1\) ↪ \(A^{1}_6\)
14 out of 61
Computations done by the calculator project.

Subalgebra type: \(\displaystyle A^{56}_1\) (click on type for detailed printout).
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer: \(\displaystyle A^{1}_6\)

Elements Cartan subalgebra scaled to act by two by components: \(\displaystyle A^{56}_1\): (6, 10, 12, 12, 10, 6): 112
Dimension of subalgebra generated by predefined or computed generators: 3.
Negative simple generators: \(\displaystyle g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-6}\)
Positive simple generators: \(\displaystyle 6g_{6}+10g_{5}+12g_{4}+12g_{3}+10g_{2}+6g_{1}\)
Cartan symmetric matrix: \(\displaystyle \begin{pmatrix}1/28\\ \end{pmatrix}\)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): \(\displaystyle \begin{pmatrix}112\\ \end{pmatrix}\)
Decomposition of ambient Lie algebra: \(\displaystyle V_{12\omega_{1}}\oplus V_{10\omega_{1}}\oplus V_{8\omega_{1}}\oplus V_{6\omega_{1}}\oplus V_{4\omega_{1}}\oplus V_{2\omega_{1}}\)
In the table below we indicate the highest weight vectors of the decomposition of the ambient Lie algebra as a module over the semisimple part. The second row indicates weights of the highest weight vectors relative to the Cartan of the semisimple subalgebra.
Highest vectors of representations (total 6) ; the vectors are over the primal subalgebra.\(g_{6}+5/3g_{5}+2g_{4}+2g_{3}+5/3g_{2}+g_{1}\)\(g_{11}+2g_{10}+12/5g_{9}+2g_{8}+g_{7}\)\(g_{15}+2g_{14}+2g_{13}+g_{12}\)\(g_{18}+5/3g_{17}+g_{16}\)\(g_{20}+g_{19}\)\(g_{21}\)
weight\(2\omega_{1}\)\(4\omega_{1}\)\(6\omega_{1}\)\(8\omega_{1}\)\(10\omega_{1}\)\(12\omega_{1}\)
Isotypic module decomposition over primal subalgebra (total 6 isotypic components).
Isotypical components + highest weight\(\displaystyle V_{2\omega_{1}} \) → (2)\(\displaystyle V_{4\omega_{1}} \) → (4)\(\displaystyle V_{6\omega_{1}} \) → (6)\(\displaystyle V_{8\omega_{1}} \) → (8)\(\displaystyle V_{10\omega_{1}} \) → (10)\(\displaystyle V_{12\omega_{1}} \) → (12)
Module label \(W_{1}\)\(W_{2}\)\(W_{3}\)\(W_{4}\)\(W_{5}\)\(W_{6}\)
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Semisimple subalgebra component.
\(-g_{6}-5/3g_{5}-2g_{4}-2g_{3}-5/3g_{2}-g_{1}\)
\(h_{6}+5/3h_{5}+2h_{4}+2h_{3}+5/3h_{2}+h_{1}\)
\(1/3g_{-1}+1/3g_{-2}+1/3g_{-3}+1/3g_{-4}+1/3g_{-5}+1/3g_{-6}\)
\(g_{11}+2g_{10}+12/5g_{9}+2g_{8}+g_{7}\)
\(g_{6}+g_{5}+2/5g_{4}-2/5g_{3}-g_{2}-g_{1}\)
\(-h_{6}-h_{5}-2/5h_{4}+2/5h_{3}+h_{2}+h_{1}\)
\(g_{-1}+3/5g_{-2}+1/5g_{-3}-1/5g_{-4}-3/5g_{-5}-g_{-6}\)
\(2/5g_{-7}+2/5g_{-8}+2/5g_{-9}+2/5g_{-10}+2/5g_{-11}\)
\(g_{15}+2g_{14}+2g_{13}+g_{12}\)
\(g_{11}+g_{10}-g_{8}-g_{7}\)
\(g_{6}-g_{4}-g_{3}+g_{1}\)
\(-h_{6}+h_{4}+h_{3}-h_{1}\)
\(-2g_{-1}+g_{-3}+g_{-4}-2g_{-6}\)
\(-2g_{-7}-g_{-8}+g_{-10}+2g_{-11}\)
\(-g_{-12}-g_{-13}-g_{-14}-g_{-15}\)
\(g_{18}+5/3g_{17}+g_{16}\)
\(g_{15}+2/3g_{14}-2/3g_{13}-g_{12}\)
\(g_{11}-1/3g_{10}-4/3g_{9}-1/3g_{8}+g_{7}\)
\(g_{6}-4/3g_{5}-g_{4}+g_{3}+4/3g_{2}-g_{1}\)
\(-h_{6}+4/3h_{5}+h_{4}-h_{3}-4/3h_{2}+h_{1}\)
\(10/3g_{-1}-8/3g_{-2}-5/3g_{-3}+5/3g_{-4}+8/3g_{-5}-10/3g_{-6}\)
\(6g_{-7}-g_{-8}-10/3g_{-9}-g_{-10}+6g_{-11}\)
\(7g_{-12}+7/3g_{-13}-7/3g_{-14}-7g_{-15}\)
\(14/3g_{-16}+14/3g_{-17}+14/3g_{-18}\)
\(g_{20}+g_{19}\)
\(g_{18}-g_{16}\)
\(g_{15}-g_{14}-g_{13}+g_{12}\)
\(g_{11}-2g_{10}+2g_{8}-g_{7}\)
\(g_{6}-3g_{5}+2g_{4}+2g_{3}-3g_{2}+g_{1}\)
\(-h_{6}+3h_{5}-2h_{4}-2h_{3}+3h_{2}-h_{1}\)
\(-5g_{-1}+9g_{-2}-5g_{-3}-5g_{-4}+9g_{-5}-5g_{-6}\)
\(-14g_{-7}+14g_{-8}-14g_{-10}+14g_{-11}\)
\(-28g_{-12}+14g_{-13}+14g_{-14}-28g_{-15}\)
\(-42g_{-16}+42g_{-18}\)
\(-42g_{-19}-42g_{-20}\)
\(g_{21}\)
\(g_{20}-g_{19}\)
\(g_{18}-2g_{17}+g_{16}\)
\(g_{15}-3g_{14}+3g_{13}-g_{12}\)
\(g_{11}-4g_{10}+6g_{9}-4g_{8}+g_{7}\)
\(g_{6}-5g_{5}+10g_{4}-10g_{3}+5g_{2}-g_{1}\)
\(-h_{6}+5h_{5}-10h_{4}+10h_{3}-5h_{2}+h_{1}\)
\(7g_{-1}-21g_{-2}+35g_{-3}-35g_{-4}+21g_{-5}-7g_{-6}\)
\(28g_{-7}-56g_{-8}+70g_{-9}-56g_{-10}+28g_{-11}\)
\(84g_{-12}-126g_{-13}+126g_{-14}-84g_{-15}\)
\(210g_{-16}-252g_{-17}+210g_{-18}\)
\(462g_{-19}-462g_{-20}\)
\(924g_{-21}\)
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)		\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)		\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)		\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)		\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)		\(12\omega_{1}\)
\(10\omega_{1}\)
\(8\omega_{1}\)
\(6\omega_{1}\)
\(4\omega_{1}\)
\(2\omega_{1}\)
\(0\)
\(-2\omega_{1}\)
\(-4\omega_{1}\)
\(-6\omega_{1}\)
\(-8\omega_{1}\)
\(-10\omega_{1}\)
\(-12\omega_{1}\)
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)		\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)		\(\displaystyle M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}
\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}\)
Isotypic character\(\displaystyle M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\)\(\displaystyle M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\)\(\displaystyle M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\)\(\displaystyle M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}
\oplus M_{-8\omega_{1}}\)		\(\displaystyle M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}\oplus M_{-4\omega_{1}}
\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\)		\(\displaystyle M_{12\omega_{1}}\oplus M_{10\omega_{1}}\oplus M_{8\omega_{1}}\oplus M_{6\omega_{1}}\oplus M_{4\omega_{1}}\oplus M_{2\omega_{1}}\oplus M_{0}\oplus M_{-2\omega_{1}}
\oplus M_{-4\omega_{1}}\oplus M_{-6\omega_{1}}\oplus M_{-8\omega_{1}}\oplus M_{-10\omega_{1}}\oplus M_{-12\omega_{1}}\)

Semisimple subalgebra: W_{1}
Centralizer extension: 0


Made total 128928 arithmetic operations while solving the Serre relations polynomial system.